Dear Michael, Thank you for your "trying" to answer. I have waited for a few days for a "real" answer from Peter Johnstone to whom my question was, for obvious reasons, primarily addressed. When my mail to you was completed I received a very complete and nice answer from Ross Street, and then, two more by Marta Bunge. I want to thank them, and tell them that I shall try to give, in detail, complete answers to their mails.That is, if the "higher authorities" who control this list consider that this "mathematico- historical" disicussion could be as important as, say, the more than 20 mails devoted to the over all important discussion about role versus r\ole. I shall wait a week before answering Ross and Marta, in case I get more answers, and, who knows, one can always dream, one by "Peter Johnstone himself". I shall make do with your answer, but before I make a few comments about it, in order for you to understand them, it would be better to read carefully my comments on the "non-answer" by Johnstone to a question which concerned his book; After all "he" was responsible for not mentioning a 1970 "Comptes Rendus" note, very frequently referred to, and attributing the result to John Beck, without any reference to a published, or unpublished, paper of his. Not even to a paper of another author, dating of before 1970, and crediting John Beck with precisely, I insist on it, the same theorem that was given in my joint note with Roubaud, and which is attributed to Beck in "The Elephant"! He is no "baby in the woods", and if he writes something in an important book, published by Oxford University Press, he must be able to explain his decision. I hope to hear from him soon, and; since I am on this unpleasant matter, I hope he shall also answer the following questions: (i) In the long Appendix of his Topos Theory (TT), there is only one theorem.It is due to a student of mine, Jean Celeyrette, whose thesis is mentioned in the bibliography of (TT). Why has the name of J.Geleyrette totally disappeared from the much much longer bibliography of (El)? (ii) Same question about my Louvain paper on "Distributors" or "Profunctors", which he uses in an essential manner in (El) without ever mentioning my name. It was also in the bibligraphy of (TT) and again absent from the bibliography of (El) With the note on descent, and many other examples, this is getting to be "a habit" with Peter Johnstone. I advise him to lose very quickly such habits, they might become dangerous for one's health. ------------------------------------------------------------------------ - Considering what I said about Johnstone, you won't be surprised if I tell you that your answer does not fully satisfy me (and that is an understatement) Since oblique, bold, etc fonts are not accepted in this list, I shall write between quotation marks any parts of your mail I want to comment upon, and without quotation marks my comments. (i) "I certainly heard Jon lecture on this a number of times" What does your "this" precisely refer to? (ii) " PUblication? Lot's of luck. A quick glance at MathSciNet shows that there are an awful lot of J. Beck's, at least one J.M. Beck and at least one Jonathan Beck, but no paper by our Jon Beck on descent theory. As for precise statement, don't even think about it". I cannot, and will not, take this for an answer. My joint theorem with Roubaud is a precise statement, and so is its reformulation by Johnstone. He was obviously too young in 1970 to have heard it, if was the same, directly from Beck. How can he be sure it was the same, if you are not? Moreover I am vain enough to consider it was an important result, because it established a connection between two important theories, namely: descent and triples. There were enough good mathematicians in North America in the late 60's, and certainly, at least one of them, would have grasped its importance, and given one, or many, applications, as we did, Roubaud and myself, in the same note where we stated the theorem. Where are these applications? (iii) "But my recollection was only whether a triple could descend across a functor. There were cocyle conditions that were necessary and sufficient. I think the "Beck-Chevalley condition" was a simple example" I do not merely "think", I am sure, that I learnt Chevalley's condition, from Chevalley, in 1964. At that time fibered categories, "invented" by Grothendieck, were almost "unheard of" in the "North American" category community. The first reference I know of is Gray's paper in the 1965 conference of La Jolla, where he refers to Chevalley's lost notes for 1962 lectures at Berkeley. Even now, they are often presented in terms of "indexed categories", under the influence of W.V. Lawvere's 1971 "Perugia Notes". I thus doubt very much that, whatever Beck's talent,in 1964, when his PhD thesis was not yet completed, he might have had anything like Chevalley's condition for arbitrary fibrations. I "think" I was wrong to "compromise" and to accept that what I called the Chevalley Condifion should have Beck's name assocoated to it, and I'm sure that, from now on, I shall call Chevalley condition what was up to now called Beck-Chevalley condition, only because I insisted that it was historically a nonsense to call it, as the North American school did, "Beck" condition ! (iii) "At one point, Jon told my wife with some regret that, thanks to my insistence, he was finally published. He seemed constitutionally incapable of putting his thoughts in public." When and where was he "finally published"? (iv) "I think you can suppose that if PTJ couldn't find it, it isn't there except in the (increasingly feeble) memories of those who heard him." I do not merely "think", I know that you are a mathematician, (and that of course for me means a good one). Thus, if Beck's formulation had been so blatantly simple and precise as mine and Roubaud's you wouldn't need an effort of memory to remember it, with precision. And this is of course also true for many of the mathematicians "who heard him". Although I was not among the happy few who heard him, I don't need a great effort of memory to remember Beck' Triplability Theorem. I "think" also that Johnstone had better find a more credible justification than mere "hear so" and "think that", Jean Benabou, out of solidarity with the so-called "category-community" might not, even if he is angry, rise such a fuss. But Jacques Roubaud has no such solidarity, is very angry, and he is known, and respected, in "circles" much wider that the few handful of persons that some of us tend to "think of" as the center of the world. I do not "think", we are the center of the world, I am even sure, we are not !